Question A.3: Rectangular-to-Polar Conversion Convert Z5 = 10 + j5 and Z6 ...
Rectangular-to-Polar Conversion
Convert Z5 = 10 + j5 and Z6 = -10 + j5 to polar form.
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The complex numbers are illustrated in Figure A.4. First, we use Equation A.1 to find the magnitudes of each of the numbers. Thus,
\left|Z\right|^2=x^2+y^2 (A.1)
\left|Z_5\right| =\sqrt{x^2_5+y^2_5}=\sqrt{10^2+5^2}=11.18
and
\left|Z_6\right| =\sqrt{x^2_6+y^2_6}=\sqrt{(-10)^2+5^2}=11.18
To find the angles, we use Equation A.2.
\tan{(\theta)}=\frac{y}{x} (A.2)
\tan{(\theta_5)}=\frac{y_5}{x_5}=\frac{5}{10}=0.5
Taking the arctangent of both sides, we have
\theta_5=\arctan{(0.5)}=26.57^\circ
Thus, we can write
Z_5=10+j5=11.18\angle 26.57^\circ
This is illustrated in FigureA.4.
Evaluating EquationA.2 for Z6, we have
\tan{(\theta_6)}=\frac{y_6}{x_6}=\frac{5}{-10}=-0.5
Now if we take the arctan of both sides, we obtain
\theta_6=-26.57^\circ
However, Z6 = -10+j5 is shown in Figure A.4. Clearly, the value that we have found for θ5 is incorrect. The reason for this is that the arctangent function is multivalued. The value actually given by most calculators or computer programs is the principal value. If the number falls to the left of the imaginary axis (i.e., if the real part is negative), we must add (or subtract) 180 to arctan(y/x) to obtain the correct angle. Thus, the true angle for Z6 is
\theta_6=180+\arctan{\left(\frac{y_6}{x_6}\right)}=180-26.57=153.43^\circ
Finally, we can write
Z_6=-10+j5=11.18\angle 153.43^\circ
